Theorem diasslssN

Description: The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	diasslss.h	|- H = ( LHyp ` K )
	diasslss.u	|- U = ( ( DVecA ` K ) ` W )
	diasslss.i	|- I = ( ( DIsoA ` K ) ` W )
	diasslss.s	|- S = ( LSubSp ` U )
Assertion	diasslssN	|- ( ( K e. HL /\ W e. H ) -> ran I C_ S )

Proof

Step	Hyp	Ref	Expression
1		diasslss.h	|- H = ( LHyp ` K )
2		diasslss.u	|- U = ( ( DVecA ` K ) ` W )
3		diasslss.i	|- I = ( ( DIsoA ` K ) ` W )
4		diasslss.s	|- S = ( LSubSp ` U )
5	1 3	diaf11N	|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )
6		f1ocnvfv2	|- ( ( I : dom I -1-1-onto-> ran I /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) = x )
7	5 6	sylan	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) = x )
8	1 3	diacnvclN	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( `' I ` x ) e. dom I )
9		eqid	|- ( Base ` K ) = ( Base ` K )
10		eqid	|- ( le ` K ) = ( le ` K )
11	9 10 1 3	diaeldm	|- ( ( K e. HL /\ W e. H ) -> ( ( `' I ` x ) e. dom I <-> ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) ) )
12	11	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( ( `' I ` x ) e. dom I <-> ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) ) )
13	8 12	mpbid	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) )
14	9 10 1 2 3 4	dialss	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) ) -> ( I ` ( `' I ` x ) ) e. S )
15	13 14	syldan	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) e. S )
16	7 15	eqeltrrd	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> x e. S )
17	16	ex	|- ( ( K e. HL /\ W e. H ) -> ( x e. ran I -> x e. S ) )
18	17	ssrdv	|- ( ( K e. HL /\ W e. H ) -> ran I C_ S )